Use the given values to evaluate(if possible) three trigonometric functions cos x, csc, tan x

Question

I must use the given values to evaluate(if possible) three trigonometric functions cos x, csc x, tan x.

Sin=1/3 cos>0

A. Cos x= -2√2/3

. Csc x=3

. Tan x= √2/4

Answer 1

given Sin x=1/3

sin x = opp / hyp = 1/3

so in right angle triangle

using pythagorean theorem

3² = 1² + adj²

adj² =8

adj= ±2√2

as cos x >0

cos x = adj / hyp = 2√2 / 3

csc x = 1/sin x = 1/ 1/3 = 3

tan x = sin x /cos x = 1/3 / 2√2 / 3

= 1/2√2

= √2 / 4

Answer 2

So, this is generally asking about the relationship between trigonometric functions. Let's summarize some known laws about trig functions:

Sin(x) / Cos(x) = Tan(x)

1 / Sin(x) = Csc(x)

1/ Cos(x) = Sec(x)

1/Tan(x) = Cot(x)

Also,

Sin(x)^2+Cos(x)^2 = 1

If I understand the problem you've written correctly we know that Sin(x) = 1/3, thus we can conclude Csc(x) = 3 immediately

Next, we know that Sin(x)^2+Cos(x)^2 = 1, so Cos(x) = sqrt(1-(1/9)). This has two possible values, 2*sqrt(2)/3 and -2*sqrt(2)/3. The problem tells us that cos>0, so the answer must be the positive 2*sqrt(2)/3

Finally for tan(x) we simply divide Sin(x) by Cos(x) and get (1/3)/(2*sqrt(2)/3) = 1/(2*sqrt(2)) = sqrt(2)/4

It looks like you have the correct answers written into your question as well.

Is there something more you're asking about?